Mathematical surprises

I’m interested in compiling a list of “mathematical surprises.” The best possible example would be a mathematical discovery that no mathematician saw coming, but after it was discovered it changed mathematics in some fundamental way—Cantor’s discovery of the nondenumerability of the continuum is such an example. But I’ll settle for any surprise—Andrew Wiles surprised everyone with his proof of Fermat’s Last Theorem, the solution of the Monty Hall problem surprised many capable mathematicians, etc.

I’ve spent a couple days brainstorming and I’ve come up with the following list. Some are better than others, and they’re listed in no particular order. Please add your surprises in the comments below!

Ito’s lemma for sure! The fact that random/stochastic behaviour turns into deterministic behaviour under certain well defined circumstances and that you have to use the second derivatives term to integrate an stochastic process – WOW!!!

This might be of interest to you: I am just reading “Darf ich Zahlen?” from the well known mathematician Günther Ziegler (TU Berlin). He has a seperate chapter on surprises (“Über Überraschungen”, p. 187 f.) There he writes about the Göttinger mathematics-philosopher Felix Mühlhölzer who has worked out a scheme of mathematical surprises on the basis of “Bemerkungen über die Grundlagen der Mathematik” from Ludwig Wittgenstein. Basically he differentiates between “R-Überraschungen” and “F-Überraschungen”: R stands for Repräsentation, so that the surprise is only based on the representation. F stands for Fakt, so that the fact itself is a surprise. Wittgenstein says that real F-Überraschungen shouldn’t exist in mathematics. Perhaps this is a good starting point for further investigations and some ordering scheme…

Addendum:
It is actually “Günter” (without the extra “h”) Ziegler.
And the article from Mühlhölzer seems to be in English: “Wittgenstein and Surprises in Mathematics”, in: Wittgenstein and the Future of Philosophy: A Reassessment after 50 Years (Proceedings of the 24th International Wittgenstein-Symposium, Kirchberg am Wechsel, 2001), hg. v. Rudolf Haller and Klaus Puhl, öbv&hpt Verlagsgesellschaft, 2002, S. 306-315.

The nonexistence of a pair of 6×6 Latin Squares. (Euler was proven correct and it only took around 100 years.) However, he was wrong when it was proved that if n>6, n=2k, and 2 doesn’t divide k, then there is a pair of orthogonal Latin squares of order n. (It only took 178 years to prove him wrong!!)

By: Barry on August 22, 2010 at 9:56 pm

Euler relation: e^(i*pi)+1=0

Does aperiodic tilings include quasiperiodic tilings? Otherwise I’d add the Penrose rhombs or kites and darts.

A fantastic list. My personal favourite is Goodstein. Getting back to basics… my eight-year son finds it really surprising that the product of two negative numbers is positive. And despite my best efforts to explain why I’m not sure he entirely believes me…

By: Jon Hinton on August 27, 2010 at 6:44 am

a few more topological surprises:
* milnor’s construction of exotic 7-spheres
* donaldson’s theorem on 4-manifolds, leading to…
* exotic R^4’s (homeomorphic but not diffeomorphic to standard R^4)
* the proof of infinitely many primes using only point-set topology (proof by furstenberg)
* all the borsuk-ulam type theorems

back down to earth:
* rearrangements of divergent series into anything you want

Euclid’s axiomatization of plane geometry and the resulting deductive system, which still serves as a paradigm for almost all of mathematics (and a great deal of physics), even though Godel’s astounding results dumped Russell and Whitehead’s “Principia Mathematica” into the garbage bin.

By: fred pollack on November 4, 2010 at 9:12 pm

[…] I was a lunch table discussion leader. Since my mathematical surprises blog post was so popular, I chose that as my table’s discussion topic. I think it went really […]

Fermat’s Last Theorem can be proved by recognising that for n greater than 2, the binomial expansion of (p+q)^n-(p-q)^n can only have an nth root if p=+q or -q.

By: Peter L. Griffiths on June 30, 2012 at 12:33 pm

Further to my comment of 30 June 2012, for n=2 the Pythagorean Triples can be easily identified from finding that pq has an integer square root.

By: Peter L. Griffiths on July 13, 2012 at 1:14 pm

A little known but very important trigonometric equation is the half angle equation cotu+cosecu=cot u/2. For those still stuck on sines and cosines, sinu=cos(90-u), tanu=cot(90-u) and secu=cosec(90-u).

By: Peter L. Griffiths on July 19, 2012 at 10:28 am

Some very advanced mathematicians do not seem to know how to compute the two square roots of the imaginary number i, or even that there are two square roots.

By: Peter L. Griffiths on July 23, 2012 at 12:00 pm

The two square roots of the imaginary number i are cos45+isin45,
and cos 225+isin225.

By: Peter L. Griffiths on December 13, 2012 at 2:54 pm

For the next lesson, what exactly are the two square roots of the imaginary number -i ?

By: Peter L. Griffiths on January 28, 2013 at 12:38 pm

Euler appears to have solved the Basel Problem by applying the Newtonian formulae to the infinite series for sines. What is not generally known is that the infinite series for cosines can be similarily used to arrive at the appropriate formula for cosines being[(PI)^2]/8=1+1/3^2 +1/5^2……

By: Peter L. Griffiths on March 17, 2013 at 2:05 pm

Further to my comment of 17 March 2013, a crucial question is where exactly did Isaac Newton first state these Newtonian formulae, the answer is to be found in vol 5 pages 358-359 of D.T. Whiteside’s Mathematical Papers of Isaac Newton. Converting infinite product series into infinite summation series and vice versa seems to be a very rare skill.

By: Peter L. Griffiths on March 23, 2013 at 2:31 pm

Can you prove the following, tan6=(tan12) (tan24) (tan48).

By: Peter L. Griffiths. on May 6, 2013 at 10:56 am

Two crucial formulae are the sinking fund formula s = (1 +r)[(1+r)^n -1]/r and the present value formula p =[1 -{1/(1 +r)^n}]/r. Both are obtained from the difference between two infinite series involving wonderful mathematics apparently too difficult for schools and universities.

By: Peter L. Griffiths on July 28, 2013 at 9:30 am

Further to my comment of 30 June 2012, I have proved Fermat’s Last Theorem in just over 400 words distinguishing between rational and irrational numbers. The nth root of 2 with n an integer is always irrational, but this irrationality can be corrected in the binomial expansion unless p and q are unequal, thus proving FLT.

By: Peter L. Griffiths on August 2, 2013 at 12:26 pm

On to another subject, The Riemann Hypothesis, Some Doubts.
Near the beginning of his 1859 paper Riemann incorrectly assumes that the complex variable s =(1/2) + ti is a zeta power. Riemann fails to recognise that an expression containing an imaginary number such as (1/2) +ti cannot be a power unless the base is a log base such as e, and also unless t being the coeffiicient of i is a specific angle. The best known example of this is Cotes’s formula cosu + isinu = e^(iu) where u is a specific angle, and it is not possible for e to be replaced by other values,
also e^(1/2) X e^(iu) equals e^[(1/2) +iu]. This means that Riemann is badly wrong in applying as a power s =(1/2) + ti. It also means that practically all the arguments in his 1859 paper are fallacious.

By: Peter L. Griffiths on December 2, 2013 at 1:30 pm

On to yet another subject, 2014 is the 400th anniversary of the discovery of logarithms by Napier probably the most important of all the mathematical discoveries, but present day mathematical societies are curiously reticent about commemorating. Is this because modern mathematicians have no idea how Napier achieved this discovery?

By: Peter L. Griffiths on January 16, 2014 at 2:54 pm

Further to my comment of 16 January 2014, it seems that Napier knew how to prove that sine 75 degrees being 0.9659258 to the power of 10 equals sine 45 degrees, can anyone else prove this, at the moment I can’t.

By: Peter L. Griffiths on January 23, 2014 at 2:53 pm

Further to my comment of 23 January 2014, the way to prove that sine 75 degrees to the power of 10 equals sine 45 degrees is to express sine 75 degrees as sine (45 + 30) which can be expanded to equal (3^[0.5] +1)/(2 X 2^[0.5]) which can be raised to the power of 10 to equal sine 45 degrees which is 0.70703.

By: Peter L. Griffiths on January 24, 2014 at 1:20 pm

Further to my comments of 23 and 24 January 2014, Napier and Regiomontanus before him knew the basic formula for constructing sine and cosine tables, this is sin2u =2sinu.cosu, which can be expressed as sin2u =2sinu.(1- [sinu]^2). This formula can be applied so that Sin30 degrees which is 1/2 which can be bisected to achieve by quadratic equations sin15 degrees which is the cosine of 75 degrees, from which sin 75 can be calculated.

By: Peter L. Griffiths on February 11, 2014 at 12:27 pm

One little known work by Johannes Kepler is the paper Concerning Conic Sections included in his book on Optics published in 1604. In this paper Kepler brings together the 5 conic sections known to the ancient Greeks, the straight line, the circle, the ellipse, the hyperbola, and the parabola and imagines wrongly that they possess in common the focus which he invented and used with pins and thread. It was not until 1618 that Kepler recognised that the common focus was the location of the Sun in relation to the orbiting planets.

By: Peter L. Griffiths on July 15, 2014 at 12:25 pm

Further to my comment of 2 August 2013, let p/q equal r which is rational, so that irrational 2^(1 -[1/n]) which contains 1/r will be reduced by a rational amount leaving another irrational amount, thus confirming Fermat’s Last Theorem.